08 - Calculate Sin, Cos & Tan w/ Unit Circle in Radians - Part 1 - Free Educational videos for Students in K-12

08 - Calculate Sin, Cos & Tan w/ Unit Circle in Radians - Part 1 - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is finding sine cosine tangent with the unit circle uh
00:06	in radiance . This is part one . So here
00:08	we put all of the skills together . We have
00:10	introduced the unit circle and degrees . We just now
00:13	in the last lesson then introduced the unit circle and
00:15	radiant measures . We now know all about radiance and
00:18	now we need to go and uh learn how to
00:21	calculate the sign , the co sign the tangent .
00:23	Also the other trig functions , the co tangent to
00:25	seek at the coast second with any angle in any
00:28	quadrant of the unit circle . So it gets tricky
00:31	and radiance because we're not comfortable yet with radiant .
00:33	So here's how we're gonna do this . We're gonna
00:35	look at the unit circle really quickly re familiarize ourselves
00:38	with it . And then as we solve the problems
00:40	I'm going to try not to use the unit circle
00:42	too much . We will use it . I will
00:45	bounce back and forth between using it and not using
00:48	it so that you can kind of check yourself and
00:49	make sure that you understand . But I really do
00:51	want to show you how to solve these problems without
00:53	using the crutch of the unit circle . It is
00:55	important to use it but it isn't important to use
00:58	it for every single problem . So the process will
01:00	work like this for every problem we're going to identify
01:03	where the angle is in radiance that's going to be
01:05	given to you . And then we have to go
01:07	figure out what quadrant that's in . And we're going
01:09	to do that by counting around the unit circle and
01:11	figuring out where our angle is . Once we know
01:14	where the angle is then we will then go figure
01:17	out the reference angle , the angle between that ray
01:21	and the nearest X . Axis . Because the sign
01:24	and the coastline is all going to be dependent upon
01:26	what the angle is between the ray that we have
01:29	, the angle that we have and the nearest X
01:30	. Axis . So once we have The sign or
01:33	the coastline or whatever we're trying to find from quadrant
01:35	one from that reference angle as if it were in
01:38	quadrant one , then we will just apply signs either
01:41	negative or positive to the answer . So I know
01:43	it sounds really complicated but it's the same exact process
01:45	. We used in degrees . Now we have the
01:47	extra wrinkle of having to deal with it radiance .
01:50	So here's the unit circle and radiance and all of
01:52	its glory . We've already gone over this extensively in
01:55	the last lesson so I'm not going to review all
01:57	of it again . What I want to focus your
01:59	attention on is quadrant one . Right here we have
02:02	zero degrees 30 degrees , 45 degrees 60 degrees and
02:05	90 degrees . The quadrant one is the most important
02:07	quadrant . Everything else comes from quadrant one . What
02:11	you need to really remember is that over here at
02:13	the zero degree line , it's zero radium . The
02:16	30 degree line is pi over 6 , 45 degrees
02:19	pi over 4 , 60 degrees pi over three and
02:21	90 degrees pi over two . We already went through
02:24	in the last lesson . Good ways to remember that
02:26	. So I'm not going to review it all again
02:28	, but basically we need to remember these guys so
02:30	that when we go in other quadrants of the unit
02:32	circle , we can easily count and figure out where
02:34	they are . Now . It's going to be much
02:37	much easier to do this by solving a problem .
02:40	So let's say we want to now find , tell
02:43	me what the sine of pi over four is .
02:47	Now . When you see signs of pie , before
02:49	there's no degree symbol or anything . And it involves
02:52	pie . Then you automatically know its radiance . You
02:55	know when you first learn uh math and trig .
02:57	You work in degrees a lot and and everybody's comfortable
03:00	with degrees . But very soon when you go up
03:02	in Matthew get away from degrees . And really you
03:05	kind of start assuming things are not in degrees anymore
03:07	. Most math beyond basic triangles and things like this
03:11	really aren't working in degrees anymore . So all of
03:14	these , especially if you see a pie involved ,
03:16	there always gonna be radiant measure . All right ,
03:18	so what is the sign of fire before ? How
03:20	do we do this ? The way you do it
03:21	is you remember pi over four is the only one
03:24	of those angles that has a four in the bottom
03:26	. And that means it's 45 degrees . That's how
03:28	you remember it . The pi over four means it's
03:30	a 45 degree angle and we know that a 45
03:33	degree angles in quadrant one in our memory . So
03:37	if we were to kind of sketch this thing which
03:38	we do want to do , I think for every
03:40	problem , even if it's just a simple little sketch
03:43	, we're not to label anything . We know that
03:45	this pi over four angle is basically going to be
03:48	at a 45 degree line . I know that may
03:49	not be exactly right , but it's basically a pi
03:52	over four is the angle or you could say 45
03:56	degrees to help you remember it . And we should
03:58	remember from the unit circle that 45 degrees . The
04:01	sign in the coastline is the same number . It's
04:02	the square root of 2/2 . So there's no mental
04:05	gymnastics . We need to do the sine of pi
04:08	before is just simply going to be the squared of
04:10	2/2 , because it's exactly the same thing as the
04:12	sign of 45 degrees . So let's go off to
04:15	the unit circle and verify that 45 degrees is right
04:19	here on the diagonal between zero and 90 . The
04:22	co sign of this angle is squared of 2/2 ,
04:24	and the sign of this angle is also squared of
04:26	2/2 , because the projection on X is down here
04:29	, which is exactly the same as the projection on
04:31	. Why . Uh by exactly that number , square
04:34	root of 2/2 . So that's the first problem for
04:37	every problem . We're going to do it like this
04:38	, we're going to sketch where it is uh here
04:42	in the beginning and then after a while , you
04:45	probably won't have to sketch every problem , but in
04:47	the beginning we definitely want to . So three pi
04:50	over four . So , this is where the wheels
04:52	come off the train a lot of times , students
04:54	will try to look at this and see this and
04:56	look at this and just try to guess the answer
04:57	. You can't do that . You have to know
04:59	what quadrant your angle is in and know exactly in
05:02	your mind where it's at . And you need to
05:04	draw a little sketch because everybody makes mistakes . We
05:06	also we all want to make sure that we have
05:08	, you know , the correct quadrant here . So
05:10	, let's look and see how we would count through
05:12	it . All right . So , if we know
05:15	this is zero and we know this is 90 degrees
05:17	. So we know that right here between must be
05:19	pi over four . And then just like we learned
05:22	in the last section , counting by pi over four
05:24	segments means this is pi over four , this is
05:27	in two pi over four and then this is three
05:30	pi over four . So three pi over four is
05:32	the angle that we want to find and we know
05:34	that it must be at this angle right here .
05:37	So this angle is actually three pi over four .
05:41	Again , that's pi over four to pi over four
05:44	to pi before remember reduces to pi over two .
05:46	Then we have three pi over four which lands here
05:49	. So here's what you do . You know that
05:52	this is in quadrant number two and you say what's
05:53	the reference angle between this ray and this axis now
05:57	because you've drawn it and you know you're counting by
05:59	45°. . The angle between here and here is just
06:03	another 45 degree angle or another pi over four radiance
06:06	. So really what you have to do is ask
06:08	yourself , okay if the reference angle is pira four
06:11	or 45 degrees , what's the sign of that sign
06:14	up before we just did that in the previous problem
06:16	. The answer to that was the square root of
06:18	2/2 . Now we have to ask ourselves should we
06:22	put a negative sign or leave it as positive ?
06:24	You look in this quadrant and its projection onto the
06:26	Y axis would be a positive number . So this
06:29	should be a positive number as well . So the
06:31	actual answer is the positive square root of two over
06:33	to you notice how we didn't have to use the
06:36	unit circle ? We just know that the sine of
06:39	pi over four is square to to over two .
06:41	And then we look at the quadrant were in to
06:42	see if it should be positive or negative . But
06:45	as a check , we go here and say this
06:48	is pi over four . To pi over 43 pi
06:50	over four . The sign of three pi over four
06:52	is positive notice positive squared of 2/2 because the projection
06:56	goes to the positive y axis right there . All
07:01	right , So let's crank up the complexity a little
07:02	bit . I think a lot of students can get
07:04	that without any problem . Uh in this next problem
07:07	is not difficult , but we want to make sure
07:10	we can do problems of all types . So ,
07:11	what if you had the sign of negative three pi
07:16	over four and get a lot of students , you
07:19	know , when you start learning this stuff , you
07:20	don't know quite what to do if it's negative three
07:22	pie before . So let me write a pie correctly
07:25	here . So there's a pie like this . Alright
07:27	, so again , it all boils down to drawing
07:30	a picture . Do not try to do this kind
07:31	of stuff in your mind , you can look at
07:34	the unit circle , that's fine . But I'm trying
07:35	to teach you how to do things so you understand
07:37	it . So this is zero and you're going negative
07:40	angles , you have to count in chunks of pi
07:42	over four . So here's one pi over four negative
07:45	one point before here's negative to pi over four .
07:48	Here's negative three pi over four . So we now
07:50	know it's right over there so we'll draw a little
07:53	ray out at a 45 degree little angle right there
07:56	. This is negative three pi over four . As
07:59	far as its angle measure like this , that's its
08:02	angle measure . This angle measure was measured this way
08:05	, so it was positive and that one was measured
08:07	there because it was positive , so negative 35 or
08:09	four . So then we ask ourselves what is the
08:12	reference angle ? I mean it's obvious . We're counting
08:14	in chunks of five or 4 45 degrees . So
08:17	the angle between this and the nearest x axis is
08:20	45 degrees . So the actual sign of this thing
08:23	, the number is going to be the sign of
08:25	45 degrees or the sine of pi before it's the
08:27	same number is the all the other problems . It's
08:29	the square root of 2/2 . But then we have
08:32	to ask ourselves uh is this should be left positive
08:36	or should we make it negative ? So we look
08:38	at this quadrant and we say the projection onto the
08:40	y axis because it's a sign is going to be
08:43	on the negative y axis . So actually the projection
08:46	here is negative square to over to you see ,
08:48	basically , since we know the ray is here ,
08:50	the projection has to be a negative squared of two
08:52	over to the projection here is on the positive y
08:55	axis , so it has to be positive squared of
08:57	2/2 . And the projection here is on the positive
08:59	y axis , positive square to over two . In
09:01	all three cases it's the same number squared of 2/2
09:05	because this thing is relatively speaking at a 45 degree
09:08	angle all over the unit circle . So the projection
09:11	here and the projection here , and the projection here
09:13	is always squared of 2/2 . That's why we use
09:16	the reference angle to figure out what the number is
09:18	. But we have to look at the quadrant to
09:20	figure out if it's going to project onto a negative
09:22	or a positive axis . So let's check ourselves -3
09:26	Pi over four . The sign of that being negative
09:29	squared of two over to you look at your unit
09:31	circle and you realize , oh no , I don't
09:35	have any negative angles on the unit circle . I
09:37	can't check myself . This is why using the unit
09:40	circle is great in the beginning , but it limits
09:43	you , if you're just using it to just get
09:45	your answers , then you can't do anything beyond a
09:48	really basic problem because there's no negative angles here .
09:50	And if you don't know how to count by pi
09:52	over four chunks in the negative direction , you'll never
09:55	know what to do because all of these angles are
09:57	positive , right ? So what you do is you
09:59	say Well , ok , I know how to count
10:01	, right ? So here is actually even though it
10:03	says seven pi over four , this is really negative
10:05	pi over four and this is really negative to pi
10:08	over four and this is really negative three pi over
10:11	four because it turns out that negative three pi over
10:13	four is exactly the same as positive five pi over
10:16	four , Same angle in the same quadrant , just
10:19	expressed differently , right ? And that's the way it
10:21	goes , like fractions , you know ? Uh one
10:24	half is the same as five tents . They look
10:26	different , but they represent the same thing . So
10:28	negative angles and positive angles can represent the same thing
10:30	. If you don't know , having unit circle is
10:32	laid out and you're just randomly doing problems , then
10:36	uh you know , you're going to end up getting
10:37	into trouble with any kind of thing . More complicated
10:41	. All right . So let's take a look at
10:44	tangent of five pi over four . All right .
10:51	Now , again , you notice that there is no
10:54	tangent on the boundaries of the unit circle . So
10:56	what you do is you say the tangent , it
10:57	can be written always as the sine of the angle
10:59	divided by the cosine of the angle . So this
11:01	is what you do it . You literally right the
11:03	sign of five pi over four . And you divide
11:08	whatever you get there by the co sign of five
11:12	pi over four . Like this . So now it
11:16	reduces to figuring out what the sign of this angle
11:18	and what the coastline of this angle is . So
11:20	the next step is really to take a look at
11:23	what quadrant this is in because it's going to determine
11:25	everything else notice it does have a four on the
11:28	bottom . So we know it's going to be some
11:29	45° angle somewhere here . So let's count . Let's
11:33	figure out where it is . Here's zero , here's
11:35	pira for one pirate for two pira , fours three
11:39	pira , fours , four pinafores , five Pirate force
11:42	. So five pi over four is going to be
11:45	at this kind of this 45 degree angle between these
11:48	two axes right here . Five pi over four .
11:52	It's a positive angle . So it's measured like this
11:55	, that's the angle five power before . So then
11:58	we have to ask ourselves what is the reference angle
12:01	to figure out what the sign of coastline as well
12:02	. It's easy in this case , this is a
12:04	45 degree angle , the reference angle to the nearest
12:06	X axis . So we're basically trying to find the
12:09	sign and the co sign of Pie before the reference
12:12	angle , which we already know is the square root
12:14	of 2/2 . So what we do is we say
12:17	, well on the top is going to be ,
12:18	the number is going to be the sign of this
12:21	. Uh , Pirate four squared of 2/2 . And
12:24	on the bottom it's also gonna be squared of 2/2
12:27	. And now we need to put the correct signs
12:29	in place . So here is the sign of this
12:32	angle . We know the number is squared of 2/2
12:34	and it's projecting onto the negative Y . Axis of
12:37	this has to be negative . The co sign of
12:40	five power before it means this is projecting up to
12:42	the negative X axis . So in this quadrant the
12:44	co sign is going to be negative too . So
12:46	what we figured out is that the sign of 55
12:49	or four is actually negative squared of two . Over
12:51	to the number squared of two or two comes from
12:54	the fact that we're just essentially we're taking the sign
12:57	of the reference angle and then we stick a negative
12:59	on because it's projecting down here here same reference angle
13:02	. So the coastline is the same number . And
13:05	then we're projecting to the negative access . So we
13:06	get a negative but negative divided by negative gives us
13:10	a positive . So actually this divided by this is
13:13	just a positive one . That's the tangent of five
13:17	power before go into a calculator . And and and
13:19	calculate the tangent of five power before or computer .
13:22	And you'll see that the answer is positive one .
13:24	So let's check ourselves . So here was pi over
13:29	four pi over four to pi Reform reduces to this
13:31	three pie before four pi over four reduces to this
13:35	five pie before is right exactly where we said .
13:38	The co sign of five pi reports negative squared of
13:40	2/2 . And the sign of five prior before is
13:43	also negative squared of 2/2 . So sine divided by
13:46	co sign is gonna be this number divided by this
13:48	number . It gives you a positive one . It's
13:50	exactly what we have said . So I think what
13:52	I can do actually is just drag this board over
13:55	here so I can leave my unit circle kind of
13:56	out . All right . And this process we're going
14:00	to continue . Let's do something that looks difficult ,
14:03	you know ? And shuts a lot of students down
14:05	. But it's not that hard . What about the
14:07	secret of three pi over 23 pi or to ?
14:12	Well , first of all , you know when you
14:15	see seeking a lot of people don't like what is
14:16	that ? So here's what you do . You go
14:18	off to the side and you say , well this
14:20	is how I write them down . Sign co sign
14:22	tangent . Then after that comes the co tangent .
14:25	Then the second then the co secret . You just
14:27	have to remember the order . This one goes with
14:29	this one , this one goes with this one ,
14:30	this one goes to this one . So the second
14:33	this guy is actually one over the co sign .
14:36	So one over the co sign of what three pi
14:40	over two . So this whole problem reduces yeah .
14:45	To figuring out what the coastline of that angle is
14:47	. So the next thing we have to do as
14:50	we need to draw a sketch on the xy plane
14:53	of where that angle actually is . Now before we
14:57	had chunks of we had a four in the denominator
15:00	but we don't have that anymore . But so when
15:02	we had four in the denominator we knew we had
15:04	to count in chunks of pyre before . So now
15:07	that the angle does not have a four has a
15:10	two in the bottom . We know we're gonna have
15:11	to count in chunks of pi over two . And
15:13	you should remember I told you have to remember a
15:15	couple of things pi over two is just the vertical
15:18	Y axis . So what we're gonna do is say
15:20	okay here's zero , here's one pi over two .
15:23	Then again counting in chunks of 90 degrees . There's
15:25	two pi over two . Then here's three pi over
15:28	two . We never had to look at a unit
15:30	circle to know that this angle was three pi over
15:33	two . So basically it's this angle down here I
15:37	guess I can try to do it in red to
15:38	kind of show you I'll draw the line like right
15:40	next to the axis but it's it's really on top
15:42	of the axis . This is the angle right here
15:44	is three pi over two so it's pointing straight down
15:46	like this . Yeah . So let me ask you
15:49	a question . I'm trying to find the co sign
15:52	of this angle . You have to put a one
15:54	on the top . Of course we have to find
15:55	the coastline of this angle . Alright , co sign
15:57	is the projection of an angle onto the X .
16:00	Axis . Okay , so if I'm shining a light
16:03	trying to project this thing onto the X axis ,
16:04	what am I gonna get ? No projection at all
16:06	? So the co sign a three pi over two
16:09	actually turns out to be what ? Zero ? There
16:11	is no projection . So I put a zero here
16:14	. Now what is 1/0 ? You can write undefined
16:17	if you want but I generally like to write infinity
16:19	. So you can say infinity or undefined because really
16:26	both answers I would consider to be correct what's gonna
16:28	happen when we graph the second function later on .
16:31	And also the tangent function . We're going to find
16:33	that because the tangent is sine over co sign because
16:37	the second is one over something . All of these
16:40	trig functions with a fraction . With something in the
16:42	bottom , there's always gonna be some angle that will
16:44	go to make this thing go to zero . And
16:46	so you'll get an infinity somewhere . So when we
16:48	graph these later on , we'll see these things shoot
16:50	up to infinity at certain angles . And this is
16:53	one of those angles at three pi over two .
16:55	Uh You get an infinity in the second curve ,
16:58	or you can just call it undefined if you want
16:59	to basically , if you get really close to three
17:01	pi over two , it goes it shoots up or
17:03	shoots down negative , depending on which way you're going
17:06	, but it's going to infinity one way or another
17:08	. All right . How would you check this ?
17:10	Well , the unit circle doesn't have anything to do
17:13	with second , but we can certainly see that here's
17:15	pi over two . There's uh sorry pi over two
17:18	. There's two prior to that reduces to this three
17:21	pi over two right here . And we were trying
17:23	to find the co sign . So the co sign
17:25	of this is zero . That's the first number .
17:27	And so zero , we can check that part in
17:29	the in a circle . 1/0 gives you infinity .
17:31	Mhm . All right , so let's check the next
17:35	guy . Let's take a look at something very similar
17:38	. What about the coast ticket of three pi over
17:42	two ? So , it's almost exactly the same thing
17:44	. It's the same angle but a different trig function
17:46	. What is the co second ? It's one over
17:49	the sign . One over the sign of three ,
17:53	pi over two . So , we don't have to
17:55	draw a figure again , it's right here . So
17:57	let's say we want to find the sign of this
17:59	angle . The sign of this angle , the sign
18:01	is the projection onto the y axis . It's a
18:03	unit circle . So this problem boils down to figuring
18:07	out what the sign of three pi over two is
18:09	. So we don't need a new figure , we
18:10	have one right here . The figure shows 35 or
18:12	two down here . And the sign is the projection
18:15	of this onto the Y axis . So what would
18:18	the projection of this on the Y axis ? B
18:20	. Well , the entire uh kinda hypotenuse array lies
18:24	on the y axis . So all of the projection
18:26	goes on the y axis here . So what would
18:28	the projection actually be when you have a one in
18:31	the numerator here ? The sign of this angle ,
18:32	the projection , look it's on the negative y axis
18:35	down here is going to be a negative one .
18:37	So when you look at the projection , you're not
18:39	just looking for the number , you're looking for the
18:40	actual sign of it . Not the sine or cosine
18:43	, I mean the the plus or minus the negative
18:46	or positive sign of it . This is going to
18:48	be projected onto the why access because it lies on
18:51	the negative Y axis . And so it has to
18:52	be negative one down here . So when you divide
18:54	this where do you get negative one ? So the
18:58	co second of this angle turned out to be negative
19:00	one while the second of the same angle turned out
19:03	to be infinity or undefined . Now again you can't
19:05	check this too much just using the unit circle .
19:08	But you can go down to three pi over two
19:10	down here and see that the sign of this angle
19:12	is actually negative one . So that's what we put
19:16	in there . So you see the unit circle is
19:18	useful . I mean I can come down here and
19:19	say oh yeah the signs negative one you can use
19:21	it . But the problem is it becomes too much
19:23	of a crutch if you if you don't even know
19:25	the basic ones like to like three pi over two
19:28	being down here and the sign of it being negative
19:30	one then you're going to use it as so much
19:32	of a crutch that you're not going to really understand
19:34	what you're doing and anything more advanced which we will
19:37	get to things more advanced . You'll have a hard
19:39	time with . But you see the way we did
19:40	this we didn't even use the unit circle . We
19:42	just said we're gonna count by pi over two .
19:45	There's one there's two there's three pi over two .
19:47	It's down here . Okay the sign of that angle
19:49	has to be negative one because it's down here .
19:52	And so we got we got the entire solution without
19:54	even looking at the unit circle . And that's what
19:56	I'm trying to teach you . All right . So
19:59	let's see if we can do the same thing .
20:00	We did the um We did the secret of this
20:03	angle . We did the coast segment of this of
20:04	this angle . And just for giggles , let's do
20:07	the tangent of Well , I was gonna say the
20:10	same angle , but actually no , it's a different
20:12	angle to pie . Let's do the tangent of two
20:14	pi over three . So , remember the tangent is
20:18	always written as the sign of the co sign .
20:20	So , it's really the sign of two pi over
20:22	three divided by the co sign Of two pi over
20:27	three . Like this . Uh Tangelo to pi over
20:31	three . All right . So then what do we
20:32	have to do ? Obviously , we have to know
20:34	where is two pi over three in the unit circle
20:36	. You can look at the unit circle . There's
20:38	nothing wrong with that . But it kind of robs
20:40	you of the ability to visualize things . So let's
20:42	do it ourselves . We need to count in chunks
20:45	of pi over three . What is a chunk of
20:47	pi over three ? Remember with a three on the
20:50	bottom , That's a 60 degree angle . If it
20:53	was a chunk of pi over six , that would
20:54	be a 30 degree angle it goes with the opposite
20:56	number . So a pi over six is actually a
21:00	30 degree angle . A pi over three with a
21:02	three on the bottom is a 60 degree angle .
21:05	This is 90 , so 60° is about like this
21:07	. So we need to count in chunks of 60°.
21:10	. So there is one pi over three , Then
21:13	another 60° would be to pi over three and then
21:17	35 or three would be here . But 35 or
21:19	three is pie . That that makes sense . So
21:21	let's do it again . There's one pi over three
21:23	. There's 25 or three . That's the answer .
21:24	That's what we're trying to find . So then we
21:27	just now draw our picture and say it's up here
21:30	. The angle here is two pi over three again
21:35	35 or three would be here and then four or
21:37	five or three would be here and then 55 or
21:40	three would be here and then 65 or three would
21:42	be here . What is 65 or three ? It's
21:43	two pi that's why you get back into the circle
21:46	. So this is where the angle is . So
21:48	the next question you ask yourself is Because I need
21:51	to know the sign . And the coastline of this
21:52	thing is . What is the reference angle ? Well
21:54	, since I know that it was 60° chunks and
21:57	I know that I landed here then . The reference
21:59	angle is 60° 60 degrees . You can say it's
22:03	pi over three but it's basically 60 degrees . So
22:05	we need to know as far as the numbers up
22:07	here . What is the sign of 60 degrees And
22:09	what's the coastline of 60 degrees ? That's what I
22:11	want to do . So in the top , what
22:13	would be the sign of 60 degrees ? Well ,
22:14	I know the sign of 30 is one half .
22:17	So the sign of 60 is the other number .
22:18	It has to be the square root of 3/2 .
22:21	And then I say , well is the projection is
22:24	the is the sign of it correct ? It's going
22:27	to project onto the positive Y axis . So it
22:29	should be a positive . So that's correct . What
22:32	is the co sign of ? Essentially the reference angle
22:34	60 degree angle . Right . The coastline of 60
22:37	is going to be one half right . How do
22:40	I know it's one half ? Well you kind of
22:42	get familiar with these things after a while and you
22:43	realize that if the sign of 30 is one half
22:46	then the co sign of the other angle 60 degrees
22:49	is also one half . Those are things you have
22:51	to work through to start to remember but the answer
22:53	is one half . But is the is the numerical
22:56	sign of it correct ? This thing is going to
22:58	project onto the negative X . Axis . So actually
23:00	it has to have a negative sign . You see
23:03	it's easy to get these numbers but if you get
23:05	the signs wrong the whole thing is wrong . So
23:07	in this quadrant makes sense the sign is positive and
23:09	the coastline is negative . Sign is positive , coastlines
23:11	negative . So then what do we get ? Um
23:14	Just catch up to myself here . Uh We have
23:16	this guy , so we have the square root of
23:18	3/2 . And we're gonna multiply changes to multiplication and
23:22	flip negative to over one . And then you see
23:25	the twos are gonna cancel . I guess I'll just
23:27	strike through them right here . So what do we
23:29	actually get negative times ? Square root of three .
23:33	And so the answer is double check myself . Yeah
23:35	negative square to three . So the answer to the
23:38	tangent of two pi over three is negative square 23
23:40	Now I grant you we had to do some mental
23:43	gymnastics to figure out what the sign and co sign
23:45	of those angles are . But I'm trying to teach
23:47	you um by doing how you would go about doing
23:50	it . Let's check ourselves . The tangent is not
23:52	written on the unit circle , but we can certainly
23:54	check and see . Is this angle correct ? It
23:57	is two pi over three pi over three to pi
23:59	over three . Okay , this is all correct .
24:01	The sign of that . Is this number positive square
24:04	to 3/2 . That's what we wrote down here .
24:07	And the co sign of this is the first guy
24:10	, the projection onto X , which was negative one
24:12	half . Project onto negative access project onto positive .
24:16	Why access project onto negative X axis . So what
24:19	you do for them to find the tangent is sine
24:22	over cosine . This divided by this , Which is
24:25	exactly what we did . This divided by this ,
24:27	we do the math , we get -23 . That's
24:29	the final answer . Yeah . Alright . We're gonna
24:33	do one more in this lesson and then we'll wrap
24:35	it up and and do some more problems in in
24:37	subsequent lessons . Let's find something that looks really crazy
24:41	. Co tangent of 11 pi over six looks really
24:47	difficult , right ? Well first of all , what
24:49	is the co tangent ? You can go over here
24:51	and you realize the co tangent is one over the
24:53	tangent . So really it's one over the tangent of
24:58	11 pi over six . Now you can certainly do
25:02	the tangent like this . But really what I like
25:04	to do is say , well this is really one
25:06	over . This is the sign of 11 pi over
25:10	six over the co sign of 11 pi over six
25:16	. But that's all in the in the denominator there
25:18	. So when I actually do the division , I
25:20	flip and multiply . Really the co tangent is equal
25:23	to the co sign of 11 pi over six .
25:28	All divided by the sine of 11 pi over six
25:32	. Now probably won't do this too much in the
25:34	future but basically you know that tangent is sine of
25:37	a co sign . Co tangent always works out to
25:39	be co signed over sign . That's what you should
25:41	really remember . But the way you get there as
25:42	you say oh it's one of the tangent , one
25:44	of her sign of a co sign , flip and
25:45	multiply . But really going forward tangent , the sine
25:48	of a co sign coach engine is co sign over
25:51	sign . That's how you should remember it . So
25:53	we need the co sign and the sign of this
25:55	angle . So we need to figure out where this
25:57	angle is to figure it out . So here's the
25:59	X . Y . Axis and we need to count
26:02	in chunks of pi over six . What is a
26:04	chunk of five or six ? That's a 30 degree
26:06	chunk . What does a 30 degree chunk look like
26:09	? This is 90 so 30 is down here And
26:11	we need 11 of these chunks . So here we
26:14	go , one pi over six to pi over 63
26:17	pi over 645 or 655 or 665 or six notice
26:23	65 or six comes out to pie 75685 or 695
26:28	or six 10 pyro six 11 pi over six .
26:33	So it's 11 5 or six . It looks something
26:37	like this , 11 pi over six . And the
26:40	angle is measured all the way around like this notice
26:42	if I had done one more step to 12 pi
26:45	over 6 , 12/6 is too , So it's that
26:47	would be two pi . So I would get right
26:49	back around where I started . So I'm very confident
26:51	that this is the correct number . And since I'm
26:53	counting in chunks of 30 degrees , I can know
26:57	from the drawing that this reference angle has to be
26:59	30 degrees because I started at a 30 degree angle
27:02	counting around . And so when I get here I
27:04	know that this reference angle is 30 degrees . So
27:07	this whole thing boils down to figuring out what is
27:09	the co sign of that reference angle 30 degrees and
27:12	put the correct signs on it And then the sign
27:14	of 30° in all of this . Right , So
27:17	what is the co sign of 30 degrees ? What
27:20	is the coastline of 30 degrees ? Well , the
27:22	coastline of 30 degrees is the square root of 3/2
27:26	. And you can go through the mental generation sign
27:27	of 30 is one half . So the coastline of
27:29	30 must be the other angle . Now let's check
27:32	the signs down here . It's projecting onto the positive
27:35	X . Axis . So this should be positive so
27:37	I'll leave that alone . What is the sign of
27:39	the reference angle ? The sign of the reference angle
27:41	being 30 degrees sign of 30 we've been saying is
27:43	one half all along . But the projection actually land
27:47	on the negative Y . Axis . So actually this
27:50	will be negative one half down here . Okay negative
27:53	one half . So you have positive square to 3/2
27:56	over negative one half . So all we have to
27:57	do is crank through that square root of 3/2 multiplied
28:01	, flip over here it'll be negative to over one
28:05	. And so what do we get ? We get
28:06	a cancellation here And so we get negative square root
28:09	of three . So actually we get exactly the same
28:12	answer and we just double check with exactly the same
28:15	answer as we got before . So it turns out
28:17	that the tangent of this angle which is way far
28:19	away , is exactly the same thing as the co
28:21	tangent of this other angle , which is way far
28:24	away from this guy . And you cannot predict ahead
28:26	of time unless you're a human calculator , how to
28:28	do that . But what we can do is we
28:30	can say we we figured out the angle was over
28:32	here about 30 degrees , exactly 30 degrees from the
28:35	X axis 11 5 or six is right here exactly
28:39	where we predicted . Pi over six to pi over
28:41	six reduces to this three pi over six , reduces
28:44	to this four pi over six reduces to this five
28:47	pi over six . Is here six pira six reduces
28:49	to this 7.68 Pirate six reduces to this 9.6 reduces
28:54	to this 10 , 5 or six reduces to this
28:56	11 pi over six reduces is this ? And then
29:00	12 , 5 or six would go there . So
29:01	here we are . And then it was co sign
29:04	divided by sine . Co sign divided by sine .
29:07	So positive , square to 3/2 , negative one half
29:11	positive , swear to 3/2 negative one half divide .
29:13	This is the answer that you get . Believe me
29:16	, I know when you learn this the first time
29:18	it's kind of overwhelming . There's a huge circle here
29:21	, a huge , you know , unit circle with
29:23	degree markings and radiant markings and crazy square roots that
29:26	go all the way around my experience is that it
29:30	seems really hard if you try to jump into these
29:32	problems without understanding really what you're doing . Because then
29:34	it seems like you're just given like a like a
29:37	like an incomprehensible thing and you just have to go
29:39	and figure it out and just kind of your guessing
29:41	and feeling you can't do that . Every problem you
29:43	have to say what am I counting by ? What
29:45	quadrant am I in how far away in degrees and
29:48	my from the X . Axis the reference angle so
29:50	that I can figure out what the sign and coastline
29:52	are . And then I got to put the signs
29:54	on there at the end . That whole process .
29:56	I'm not dumbing it down for you . I'm not
29:58	doing it like that for you because you know you're
30:01	just learning that's exactly how I do it . If
30:03	I show you my paper that's how I'm working through
30:05	every one of these . So I'm not doing anything
30:07	on my own paper different than I would do or
30:10	tell you to do . So what I want you
30:11	to solve all of these yourself . Maybe watch it
30:14	a few times if any of these scenes you're unsure
30:17	of them and then follow me on to the next
30:19	lesson . We're gonna get a lot more practice with
30:20	this because it really is one of the most important
30:22	skills . So do these . Follow me on .
30:25	Let's get more practice with finding sine cosine tangent using
30:28	radiant measurement in the unit circle .

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